Shapes in real space ––> reciprocal space

1. Shapes in real space ––> reciprocal space (see Volkov & Svergun, J. Appl. Cryst. (2003) 36, 860-864. Uniqueness of ab initio shape determination in small-angle scattering) Can compute scattering patterns for different shape particles for isotropic dilute monodisperse systems

2. Shapes in real space ––> reciprocal space (see Volkov & Svergun, J. Appl. Cryst. (2003) 36, 860-864. Uniqueness of ab initio shape determination in small-angle scattering) Can compute scattering patterns for different shape particles for isotropic dilute monodisperse systems Approach 1 (small number of parameters) Represent particle shape by an envelope fcn – spherical harmonics

3. Shapes in real space ––> reciprocal space (see Volkov & Svergun, J. Appl. Cryst. (2003) 36, 860-864. Uniqueness of ab initio shape determination in small-angle scattering) Can compute scattering patterns for different shape particles for isotropic dilute monodisperse systems Approach 1 (small number of parameters) Represent particle shape by an envelope fcn – spherical harmonics Spherical harmonics fcns are angular part of soln to wave eqn Of the form

4. Shapes in real space ––> reciprocal space Approach 1 (small number of parameters) Spherical harmonics fcns are angular part of soln to wave eqn Of the form

5. Shapes in real space ––> reciprocal space Approach 2 (large number of parameters) Represent particle shape by assembly of beads in confined volume (sphere) Beads are either particle (X =1) or 'solvent' (X =0) To get scattered intensity:

6. Shapes in real space ––> reciprocal space bead 'annealing' envelope

7. Shapes in real space ––> reciprocal space bead 'annealing'

8. Shapes in real space ––> reciprocal space bead 'annealing' envelope

9. Shapes in real space ––> reciprocal space bead 'annealing'

10. Syndiotactic polystyrene (see Barnes, McKenna, Landes, Bubeck, & Bank, Polymer Engineering & Science (1997) 37, 1480. Morphology of syndiotactic polystyrene as examined by small angle scattering) Semicrystalline PS

11. Syndiotactic polystyrene (see Barnes, McKenna, Landes, Bubeck, & Bank, Polymer Engineering & Science (1997) 37, 1480. Morphology of syndiotactic polystyrene as examined by small angle scattering) Semicrystalline PS Expect peaks in scattering data typical of lamellar structure

12. Syndiotactic polystyrene (see Barnes, McKenna, Landes, Bubeck, & Bank, Polymer Engineering & Science (1997) 37, 1480. Morphology of syndiotactic polystyrene as examined by small angle scattering) Semicrystalline PS Expect peaks in scattering data typical of lamellar structure non-q–4 slope due to mushy interface

13. Syndiotactic polystyrene Semicrystalline PS Propose absence of peaks due to nearly identical scattering densities of amorphous & crystalline regions High temperature saxs measurements done

14. Syndiotactic polystyrene Semicrystalline PS Propose absence of peaks due to nearly identical scattering length densities of amorphous & crystalline regions High temperature saxs measurements done

15. Syndiotactic polystyrene Semicrystalline PS lamellar thickness = 18 nm averages of intensity data around azimuth

16. Syndiotactic polystyrene Semicrystalline PS